I am 42, premenopausal and have osteopenia. is it possible for me to reverse the osteopenia, perhaps by training with weights?

Osteopenia: Yes. Other treatable causes have likely been ruled out - medications (antidepressants, histamine blockers, etc), vitamin d or testosterone deficiency, hypocalcemia, substance abuse, or poor nutrition.

Yes: I wonder why you had a DEXA so young. This test is for women 60 and older usually. Strength training is a great way to "exercise" your bones and so is jogging. Frame it this way, 42 year old women are at low risk for osteoporotic fractures. Google the frax risk calculator and ask your doctor about it. This puts fracture risk in the proper perspective with other risk factors.

Expectations.: While you can attempt to stop the progression of osteopenia through weight-bearing exercise, medications, calcium and vitamin D, it's unlikely your bone strength will ever return to normal. At best, your bones may strengthen a little bit, but not back to normal levels.